An algebraic model for the free loop space

Manuel Rivera

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arXiv · 2210.10096

An algebraic model for the free loop space

Manuel Rivera

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Abstract

We describe an algebraic chain level construction that models the passage from an arbitrary topological space to its free loop space. The input of the construction is a categorical coalgebra, i.e. a curved coalgebra satisfying certain properties, and the output is a chain complex. The construction is a modified version of the coHochschild complex of a differential graded (dg) coalgebra. When applied to the chains on an arbitrary simplicial set $X$, appropriately interpreted, it yields a chain complex that is naturally quasi-isomorphic to the singular chains on the free loop space of the geometric realization of $X$. We relate this construction to a twisted tensor product model for the free loop space constructed using the adjoint action of a dg Hopf algebra model for the based loop space.

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Manuel Rivera. 2022-10-18. An algebraic model for the free loop space. https://arxiv.org/abs/2210.10096

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